University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2013

Transcription

1 Lecture 8. 1/5/13 University o Washington Department o Chemistry Chemistry 53 Winter Quarter 013 A. Cooperative Ligand inding Cooperative binding, like cooperativity in helix-coil transitions, results rom depletion o intermediates. We can see this or the extreme case o ully cooperative binding, using equation 7.9: Q= P 1+ k L + 6kk L + kkk L 3 + kkkk L (8.1) ( ) To make the transition ully cooperative, delete the weightings or singly, doubly and triply bound species to get Q = P 1+ k k k k L = P 1+ L (8.) ( 1 3 ) ( ) Now use the equation or calculating the average sites bound: ν = = ( 1+ ) = Q L 1+ L L 1+ L Equation 8.3 can be generalized to N (8.3) ν L = = (8.) N N 1 + L Equation 8. again results in a sigmoidal dependence o the saturation parameter ν on [L] as shown in Figure 8.1. Figure 8.1 saturation parameter (i.e. average number o bound sites ν ) as a unction o ligand concentration [L] or binding to a site target using the independent binding equation 7.11 and or ully cooperative behavior equation 8. and N=. 0

2 In practice equations or non-cooperative and cooperative binding are tested against data with linearized equations. For independent, non-cooperative binding the Langmuir equation (7.11) N ν = (8.5) 1 + L is linearized to ν = N ν (8.6) which is called the Scatchard equation. ν as a unction o ν yields a straight line Equation 8.6 means that a plot o [ L ] with slop and y-intercept N. Equation 8. is written as ln = ln + N ln (8.7) Equation 8.7 is called the Hill Equation and is used to it binding data or systems thought to undergo cooperative ligand binding. According to the Hill equation a plot o ln versus ln[ L ] yields a straight line with slope=n and y- 1 intercept=ln. The tetrameric O transport protein hemoglobin (Hb) has our binding sites. It is well-known that Hb s O binding data do not agree with equations 8.6 or 8.7. Non-linear scatchard plots indicate possibility o cooperative binding, but a Hill plot o oxygen binding to Hb is also nonlinear, see Figure 8.. Figure 8.: Hill plot o oxygen binding to Hb. Y=.The nonlinear Hill plot has slopes o 1 or low and high oxygen concentrations and a integer slope o.8-.9 at intermediate oxygen concentrations. 1

3 Hb data shown in Figure 8. have been itted to a number o models. An empirical equation is based on simply itting the central slope to αh PO = or ln = ln + α H ln PO (8.8) αh 1+ PO P O is the partial pressure o oxygen above the solution and α H is called the Hill parameter, which is adjusted to.8-.9 to it the intermediate slope in the Hb plot. An equation that its the ull range o the Hb Hill plot was derived by G.S. Adair in 195. It starts with equation 8.1 written as 3 Q= [ Hb]( 1+ k1po + 6k ) 1kPO + k 1kk3PO + k 1kk3kPO (8.9) We then obtain the average number o sites bound in the usual way PO 3 ν = = kp 1 O 3kkP 1 O 3kkkP 1 3 O kkkkp 1 3 O Q PO Q (8.10) Although equations 8.8 and 8.10 can it part o all o the oxygen-hb binding data they aord little insight into the basis or the cooperative binding. Figure 8.3: Juxtaposition o the concerted and sequential cooperative binding models. Sequential models propose that ligand binding induce structural conversions o low ainity binding sites to higher ainity binding sites. Concerted models impose a symmetry condition that all sites must be in the low ainity or ainity orms. The T and orms are in equilibrium in the absence o ligand where the low ainity T orm dominates. As ligand is added the equilibrium shits to the orm..as Figure 8.3 shows there are two general lines o thought about allosteric eect in Hb. o Sequential models assume addition o ligand drives structural changes that convert low ainity binding sites to high ainity binding sites. This is shown as the sequence o binding structures along the diagonal in Figure 8.3. o To quantiy a sequential binding model we start with equation 8.1 and 3 assume k1 = α k; k = α k k3 = αk; k = k where α is a parameter less than one. To simulate changes in binding site ainity, the power to which α is raised is reduced as we go rom k 1 to k.:

4 3 ( ) 1 α 3 6α 5 α 6 3 α 6 Q= Hb + k O + k O + k O + k O (8.11) o Equation 8.11 can be used to obtain the average number o sites bound as usual: 3 [ O] α k[ O] ν = = ( 1+ 3α k O + 3α k O + α k O ) (8.1) Q O Q Equation 8.1 can capture the essential eatures o the Hb Hill plot as shown in Figure 8.. In Figure 8. A Hill plot is shown or a our binding site model using equation 8.1 where α=0.5 and =100. X-ray crystallography studies o Hb and its complexes with oxygen indicated presence o several structural orms o Hb. To account or these acts, concerted models or Hb binding to oxygen were developed. Concerted Models o Hb allostery impose a symmetry requirement. The protein exists as an equilibrium between the T or tense orm and the or relaxed orm: L [ T ] T; L= (8.13) According to the symmetry requirement, all binding sites in the T state are low ainity and in the state all binding sites are high ainity. See Figure 8.3. In the absence o oxygen the T or is avored so that L>>1. As oxygen is added the equilibrium shits toward the state. The T and orms o Hb have dierent ainities or oxygen (X). For the purpose o illustrating a concerted model let us derive the binding polynomial or two oxygen binding sites N= and then generalize the result to N=. In addition to the equilibrium in equation 8.13 we have the additional equilibria T TX TX T T + X TX; TX + X TX; T = = [ T][ X] [ TX][ X] (8.1) [ ] X X + X X ; X + X X ; = = [ ][ X] [ X][ X] Note the equilibrium constants in equation 8.1 are microscopic. This act is important when we derive the binding polynomial Q. Also, because the orm binds oxygen more strongly that the T orm >> T. 3

5 Now we enumerate all the T and species into the binding polynomial which we call Q@ to remind ourselves that this is a model or N=: Q = [ T ] + [ TX ] + [ TX ] + [ ] + [ X ] + [ X ] (8.15) Now use the equilibrium expressions in equation 8.1 to eliminate the TX, X, TX and X terms. Q = [ T ] + [ TX ] + [ TX ] + [ ] + [ X ] + [ X ] (8.16) = [ T] + T [ T][ X] + T [ T][ X] + [ ] + [ ][ X] + [ ][ X] Now we use equation 8.13 to eliminate [T] rom equation = + T + T L[ ] TL[ ][ X] TL[ ][ X] [ ] [ ][ X] [ ][ X] [ ] L 1 [ X] [ X] 1 X X Q T T X T X X X ( ( T T ) ) ( ( 1 T ) ( 1 ) ) = L + X + + X = = (8.17).Equation 8.17 can now be easily generalized to our binding sites N=: Q ( ( ) ( ) ) = L 1+ T X + 1+ X (8.18) The our site concerted binding polynomial in equation 8.18 can now be used to obtain the average number o sites bound in Hb as a unction o oxygen concentration: [ X] [ ][ X] ν = = ( ) ( ) ( L 1+ T X + 1+ X ) Q X Q X 3 3 (8.19) LT ( 1+ T [ X ]) + ( 1+ [ X ]) = [ X ] L 1+ X + 1+ X ( T ) ( ) Equation 8.19 can also simulate the Hb-oxygen binding data shown in Figure 8.. To show this we consider the data in the limits o low oxygen concentration (i.e. [X]<<1) and high oxygen concentration (i.e. [X]>>1). In the limit [X]<<1, the T orm dominates so that L>>1. Then ν = [ X ] T or ln = ln T + ln[ X] (8.0) ν Thereore equation 8.0 is a Hill plot with slope =1 and y-intercept ln T. In the limit [X]>>1, the T orm dominates so that L<<1. Then ν = [ X ] or ln = ln + ln[ X] (8.1) ν Thereore equation 8.0 is a Hill plot with slope =1 and y-intercept ln. The concerted model has the appeal o explaining a great body o Hb-O binding data with a airly simple binding equation. Sequential binding equations are less general and must be written or speciic geometries. ecent data indicate the binding o oxygen by Hb has elements o both models.